In the News (Tue 18 Dec 18)

Probabilities P(E) are assigned to events E according to the probability axioms.

A Bayesian may assign a probability to the proposition that there was life on Mars a billion years ago, since that is uncertain; a frequentist would not assign such a probability, since it is not a random event that has a long-run relative frequency of occurrence.

probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions.

Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events.

Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable effect on the population as a whole, statistically.

encyclopedia.worldsearch.com /probability.htm (2610 words)

BIPS: Bayesian Inference for the Physical Sciences(Site not responding. Last check: 2007-11-05)

Probability of an event in this view is defined as the limiting frequency of occurrence of this event in an infinite number of trials.

Probability of an event in this view is determined by physical, objective properties of the object or the process generating the event.

For example, the probability of heads in a single coin toss is determined by the physical properties of the coin, such as its flat symmetric shape and its two sides.

www.sis.pitt.edu /~genie/GeNIeHelp/Probability.htm (490 words)

Bayesian probability(Site not responding. Last check: 2007-11-05)

Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements; when used with Bayes theorem, it then becomes Bayesianinference.

This is in contrast to frequentism, which rejects degree-of-belief interpretations of mathematical probability, and assigns probabilities only to random events according to their relative frequencies of occurrence.

The Bayesian approach is in contrast to the concept of frequency probability where probability is held to be derived from observed or imagined frequency distributions or proportions of populations.

www.sciencedaily.com /encyclopedia/bayesian_probability (1013 words)

Probability - Open Encyclopedia(Site not responding. Last check: 2007-11-05)

As with the theory of mechanics which assigns precise definitions to such everyday terms as work and force, so the theory of probability attempts to quantify the notion of probable.

In Cox's formulation, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions.

The probability of an event is generally represented as a real number between 0 and 1.

Bayes' theorem is a result in probabilitytheory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone.

In the context of Bayesianprobabilitytheory and statistical inference, the marginal probability distribution of A alone is usually called the prior probability distribution or simply the prior.

A Bayesian network is essentially a mechanism for automatically generating the extensions of Bayes' theorem that are appropriate for a given decomposition of the joint probability.

However this seems to be a modern meaning of the adjective Bayesian, having more to do with the use and significance of Bayes theorem in identifying belief intervals than it has to do with the use or abuse (as the case may be) of Bayesian reasoning in understanding either probabilitytheory or quantum mechanics.

Bayesian reasoning is quite different, and concerns the use of subjective probability measures in decision making processes even in the absence of a meaningful data sample.

The position I have objected to is that because probabilitytheory is ultimately Bayesian and subjective it is legitimate to modify probabilitytheory in order to fit the data, and that this constitutes an interpretation of quantum mechanics.

The basic difference between Bayesians and frequentists is this: Bayesians condition on the data actually observed, and consider the probability distribution on the hypotheses; they believe it reasonable to put probability distributions on hypotheses and they behave accordingly.

Bayesianprobability is not necessarily subjective; there is a school of objective Bayesianism, typified by people like Harold Jeffreys, the British astronomer/geophysicist/statistician, and Ed Jaynes, the physicist at Washington University in St. Louis.

Furthermore, Bayesianinference is squarely linked to the likelihood function, which even a frequentist admits has a probability interpretation (the likelihood function is proportional to the sampling distribution, which frequentists use as the basis of frequentistinference).

The original probability that a woman has cancer is so extremely low that, although a positive result on the mammography does increase the estimated probability, the probability isn't increased to certainty or even "a noticeable chance"; the probability goes from 1:1,000,000 to 1:100,000.

To increase the probability of a theory you must expose it to tests that can potentially decrease its probability; this is not just a rule for detecting would-be cheaters in the social process of science, but a consequence of Bayesianprobabilitytheory.

Expressed as a probability or a fraction, p(QandP) is the proportion of things that have property Q and property P within all things; i.e., the proportion of "women with breast cancer and a positive mammography" within the group of all women.

An approach using the Bayesianprobabilitytheory for mapping mineral potential is demonstrated in the Baguio mineral district of the Philippines.

The resulting map of posterior probabilities based on the large-scale gold occurrences shows that 79 percent of the known large-scale gold occurrences are associated with zones having posterior probabilities greater than, or equal to, the prior probability.

The resulting map of posterior probabilities based on the small-scale gold occurrences shows that 70 percent of the known small-scale gold occurrences are associated with zones having posterior probabilities equal to, or greater than, the prior probability.

Probabilitytheory is the special case of quantum mechanics in which ones algebra of observables is commutative.

Bayesian analysis also makes your life infinitely simpler, in the sense that you don't have to run around remembering a zillion different classical-statistical formulae for the case of normal distribution with known mean and unknown variance, unknown mean and known variance, and so on.

Jeffreys, _Theory of Probability_ (Oxford) Jeffreys was of the objectivist school, as is Jaynes.

A key feature of Bayesian methods is the notion of using an empirically derived probability distribution for a population parameter.

Bayesian proponents argue, correctly, that the classical methods of statistical inference have built-in subjectivity (through the choice of a sampling plan and the assumption of ‘randomness’ of distributions) and that an advantage of the Bayesian approach is that the subjectivity is made explicit [4].

Thus, the probability that the taxis the witness claimed to be blue actually being blue, given the witness's identification ability, is 12/29, i.e.

There are other domains, most notably measure theory, where the same rules appear, but from the point of view of learning systems and decisions in the face of uncertainty, degree of belief is the appropriate interpretation.

In particular, under the belief interpretation probability is not an objective property of some physical setting, but is conditional to the prior assumptions and experience of the learning system.

It is completely reasonable to talk about ``the probability that there is a tenth planet in the solar system'' although this planet either exists or does not exist and there is no sense in interpreting the probability as a frequency of observing a tenth planet.

www.cis.hut.fi /harri/thesis/valpola_thesis/node12.html (224 words)

Amazon.ca: Books: Probability Theory: The Logic of Science(Site not responding. Last check: 2007-11-05)

Probabilitytheory as a branch of mathematics makes no claim what it models.

Jaynes's advice on avoiding errors in the application of probabilitytheory -- reinforced in many examples throughout the book -- is by itself well worth the price of the book.

It convincingly shows that "statistics", "statistical inference", "Bayesianinference", "probabilitytheory", "maximum entropy methods", and "statistical mechanics" are all parts of a large coherent theory that is the unique consistent extension of logic to propositions that have degrees of plausibility attached to them.

Jaynes had essentially four different areas of research: his first could be called applied classical electrodynamics; his second, information theory (entropy as a measure of information); his third, probabilitytheory; and finally, semiclassical and neoclassical radiation theory.

This reformulation of the theory simplified the mathematics, allowed for fundamental extensions of the theory, and reinterpreted statistical mechanics as inference based on incomplete information.

He also helped take an interpretation of probabilitytheory from being virtually unknown to a healthy research area that is being applied daily in economics, biology, physics, nuclear magnetic resonance and many other disciplines.

Bayesianprobabilitytheory can be conveniently summarised in the following elementary rules:

B) denotes the probability of A on the condition that B is true.

This means that one can measure the degrees of beliefs on any scale, but it is possible to transform the degrees of beliefs on the canonical scale of probabilities such that the rules for negation and conjunction take the form of the sum and product rule.

www.cis.hut.fi /harri/thesis/valpola_thesis/node14.html (186 words)

A BAYESIAN VARIANT OF SHAFER'S COMMONALITIES FOR MODELLING UNFORESEEN EVENTS(Site not responding. Last check: 2007-11-05)

Shafer's theory of belief and the Bayesiantheory of probability are two alternative and mutually inconsistent approaches toward modelling uncertainty in artificial intelligence.

Expected utility theory requires the decision maker to assign a utility to each decision conditioned on every possible event that might occur.

Implementing this commonsensical solution is equivalent to replacing Bayesian subjective probabilities over the space of foreseen and unforeseen events by random set theoryprobabilities over the space of foreseen events.

Bayesian Knowledge Discovery home page of the Bayesian Knowledge Discovery Project, a joint effort of the Knowledge Media Institute and the Department of Statistics of The Open University - Here you can download freely the Bayesian Knowledge Discovere (BDK).

Bayesian Knowledge Discoverer Bayesian Knowledge Discovery is a Project, a joint effort of the Knowledge Media Institute and the Department of Statistics of The Open University that has produced a free tool for setting up Bayesian Networks

Rehder, B. A causal-model theory of conceptual representation and classification.

Edwin T. Jaynes was one of the first people to realize that probabilitytheory, as originated by Laplace, is a generalization of Aristotelian logic that reduces to deductive logic in the special case that our hypotheses are either true or false.

These article are on the application of probabilitytheory to the problem of estimating the frequency of oscillation of a non-sinusoidal signal in data that consists of counts.

Sivia: We have three papers by Dr. Devinder Sivia on the application of Bayesianprobabilitytheory to spectral analysis, analyzing quasielastic neutron scattering data, and extracting Structure-Factor amplitudes from powder diffraction data.

bayes.wustl.edu (773 words)

A Bayesian Theory(Site not responding. Last check: 2007-11-05)

The theory, which derives from Bayesian principles of rationality, offers a methodological framework and mathematical concepts for modeling warning systems in communities exposed to flash floods or rapid riverine floods.

At the heart of this quantification is the Bayesian processor of forecasts (BPF) which outputs a posterior description of uncertainty about flood occurrence and crest height, conditional on a flood crest forecast.

One of the formidable challenges in applying the BPF in this and other contexts has been modeling of likelihood functions and derivation, or computation, of the posterior distribution when the prior distribution is not a member of the conjugate family for a specified likelihood model.